On the Farthest Line-Segment Voronoi Diagram

The farthest line-segment Voronoi diagram shows properties surprisingly different than the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convexhull properties. In this paper we introduce the farthest line-segment hull, a cyclic structure that relates to the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull relates to the farthestpoint Voronoi diagram and provide O(n log n)-time algorithms for its construction. Using the farthest line-segment hull, we derive a more tight bound on the (linear) size of the farthest line-segment Voronoi diagram. We also illustrate properties of the L∞ farthest line-segment Voronoi diagram, which finds applications in VLSI Design Automation.